Constructing Nearfields

There are two types of finite nearfield: the regular nearfields of Dickson and the irregular nearfields of Zassenhaus. In order to accommodate both types Magma has a ‘virtual type’ Nfd and types NfdDck and NfdZss which inherit from Nfd.

Dickson Nearfields

In order to begin exploring Nfd types in Magma we need a way to create instances of nearfields and their elements. As already mentioned there is a large class of nearfields first described by L. E. Dickson [Dickson, 1905, Dickson, 1905] in 1905 and in this section we describe how to construct them in Magma.

The nearfields resulting from this construction will be called Dickson (or regular) nearfields.

If \(p\) is a prime and if the positive integers \(h\) and \(v\) satisfy

• if \(r\) is a prime or \(4\) and if \(r\) divides \(v\), then \(r\) divides \(p^h-1\)

then \((p,h,v)\) is a Dickson triple.

If we write \(q = p^h\), the condition above is equivalent to

• All prime factors of \(v\) divide \(q-1\) and \(q\equiv 3 \bmod 4\) implies \(v \not\equiv 0 \bmod 4\).

We call \((q,v)\) a Dickson pair.

DicksonPairs(p, hlo, hhi, vlo, vhi): RngIntElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> SeqEnum

The list of Dickson pairs \((q, v)\) for prime \(p\), where hlo and hhi are the lower and upper bounds on \(h\) and where vlo and vhi are the lower and upper bounds on \(v\).

DicksonPairs(p, h1, v1): RngIntElt, RngIntElt, RngIntElt -> SeqEnum

The list of Dickson pairs \((p^h, v)\) for the prime \(p\), where \(h1\) and \(v1\) are upper bounds on \(h\) and \(v\).

DicksonTriples(p, hb, vb): RngIntElt, RngIntElt, RngIntElt -> SeqEnum

The list of Dickson triples \((p,h,v)\) for the prime \(p\), where \(hb\) and \(vb\) are bounds on \(h\) and \(v\).

Example: dicksonpairs (ex-d7494c)

For each Dickson pair (equivalently Dickson triple), there is at least one Dickson nearfield.

> DicksonPairs(5,3,4,4,5);
[
    [ 125, 4 ],
    [ 625, 4 ]
]
> DicksonPairs(5,4,5);
[
    [ 5, 1 ],
    [ 5, 2 ],
    [ 5, 4 ],
    [ 25, 1 ],
    [ 25, 2 ],
    [ 25, 3 ],
    [ 25, 4 ],
    [ 125, 1 ],
    [ 125, 2 ],
    [ 125, 4 ],
    [ 625, 1 ],
    [ 625, 2 ],
    [ 625, 3 ],
    [ 625, 4 ]
]
> DicksonTriples(5,4,5);
[
    [ 5, 1, 1 ],
    [ 5, 1, 2 ],
    [ 5, 1, 4 ],
    [ 5, 2, 1 ],
    [ 5, 2, 2 ],
    [ 5, 2, 3 ],
    [ 5, 2, 4 ],
    [ 5, 3, 1 ],
    [ 5, 3, 2 ],
    [ 5, 3, 4 ],
    [ 5, 4, 1 ],
    [ 5, 4, 2 ],
    [ 5, 4, 3 ],
    [ 5, 4, 4 ]
]

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The isomorphism type of a Dickson nearfield depends on the choice of primitive element of the underlying Galois field. It has been shown by Lüneburg [Lüneburg, 1971] that if \(\phi\) is the Euler phi-function and \(g\) is the order of \(p\) modulo \(v\), there are \(\phi(v)/g\) isomorphism classes of Dickson nearfields with the same Dickson triple \((p,h,v)\).

The default nearfield will use the ‘standard’ primitive element of the field. The other variants with the same Dickson pair can be obtained by providing an integer \(s\) coprime to \(v\). Internally this is converted to a suitable integer \(e\) coprime to \(q^v -1\) such that \(s\equiv e \pmod v\).

NumberOfVariants(q, v): RngIntElt, RngIntElt -> RngIntElt

The number of non-isomorphic nearfields with Dickson pair \((q,v)\).

NumberOfVariants(N): NfdDck -> RngIntElt

The number of variants of the Dickson nearfield \(N\).

VariantRepresentatives(q, v): RngIntElt, RngIntElt -> SeqEnum

Representatives for the variant parameter of nearfields with Dickson pair \((q,v)\).

Example: variants (ex-20e6ea)

For each Dickson pair there can be several variants. The variant representative can be used when constructing the corresponding Dickson nearfield.

> NumberOfVariants(625,4);
2
> VariantRepresentatives(625,4);
[ 1, 3 ]

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DicksonNearfield(q, v : parameters): RngIntElt, RngIntElt -> NfdDck

Variant      : RngIntElt                    Default: 1
LargeMatrices: BoolElt                      Default: false

Create a Dickson nearfield from the Dickson pair \((q,v)\). The Variant parameter is an integer \(s\) which can be used to specify the choice of primitive element (see the discussion following the intrinsic DicksonTriples). The parameter LargeMatrices is used only when the group of units of the nearfield is requested. The default is to represent the group of units as a matrix group defined over the kernel of the nearfield. But if LargeMatrices is true, the matrices are defined over the prime field.

Example: dickson (ex-da0033)

As indicated in the previous example, up to isomorphism, there are two Dickson nearfields with Dickson pair \((625,4)\).

> D := DicksonNearfield(625,4);
> D3 := DicksonNearfield(625,4 : Variant := 3);
> D5 := DicksonNearfield(625,4 : Variant := 5);
> D eq D3;
false
> D3 eq D5;
false
> D eq D5;
true
> D;
Nearfield D of Dickson type defined by the pair (625, 4)
Order = 152587890625

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Zassenhaus Nearfields

It was shown by Zassenhaus [Zassenhaus, 1935] that in addition to the regular nearfields there are seven irregular nearfields. Zassenhaus gave constructions but did not prove their uniqueness. The proofs in [Zassenhaus, 1935] are known to contain gaps. Perhaps the most reliable account of the existence and uniqueness of the irregular nearfields is the PhD thesis of Dancs-Groves [Groves, 1974].

The seven finite nearfields which are not Dickson nearfields are the Zassenhaus nearfields.

Zassenhaus nearfields can be distinguished from regular nearfields by the fact that the multiplicative group of a finite nearfield \(N\) is metacyclic if and only if \(N\) is regular.

As a consequence, a Zassenhaus nearfield cannot occur as a subfield of a Dickson nearfield.

ZassenhausNearfield(n): RngIntElt -> NfdZss

Creates the \(n\)th Zassenhaus nearfield.

Example: zassenhaus (ex-af8523)

The orders of the Zassenhaus nearfields are \(5^2\), \(11^2\), \(7^2\), \(23^2\), \(11^2\), \(29^2\) and \(59^2\).

> for n := 1 to 7 do ZassenhausNearfield(n); end for;
Irregular nearfield Z with Zassenhaus number 1
Order = 25
Irregular nearfield Z with Zassenhaus number 2
Order = 121
Irregular nearfield Z with Zassenhaus number 3
Order = 49
Irregular nearfield Z with Zassenhaus number 4
Order = 529
Irregular nearfield Z with Zassenhaus number 5
Order = 121
Irregular nearfield Z with Zassenhaus number 6
Order = 841
Irregular nearfield Z with Zassenhaus number 7
Order = 3481

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